1. Field of the Invention
The present invention generally relates to a method and apparatus for controlling the operation of multiple input, multiple output (MIMO) systems. More particularly, the present invention relates to control systems for papermaking machines of the type that have actuators arranged in the cross-direction.
2. State of the Art
In the art of making paper with modern high-speed machines, sheet properties must be continually monitored and controlled to assure sheet quality and to minimize the amount of finished product that is rejected when there is an upset in the manufacturing process. The sheet variables that are most often measured include basis weight, moisture content, and caliper (i.e., thickness) of the sheets at various stages in the manufacturing process. These process variables are typically controlled by, for example, adjusting the feedstock supply rate at the beginning of the process, regulating the amount of steam applied to the paper near the middle of the process, or varying the nip pressure between calendaring rollers at the end of the process.
On-line measurements of sheet properties can be made in both the machine direction and in the cross direction. In the sheetmaking art, the term "machine direction" refers to the direction that the sheet material travels during the manufacturing process, while the term "cross direction" refers to the direction across the width of the sheet which is perpendicular to the machine direction.
Cross-directional measurements are typically made using a scanning sensor that periodically traverses back and forth across the width of the sheet material. A complete scan from one edge of the sheet to the other typically requires between twenty and thirty seconds for conventional high-speed scanners, with one measurement being read from the sensor about every fifty milliseconds. Consequently, approximately 600,000 measurements are made during each cross-directional scan.
In conventional practice, papermaking machines have several control stages with numerous, independently-controllable actuators that extend across the width of the sheet at each control stage. For example, a papermaking machine might include a steam box having numerous steam actuators that control the amount of heat applied to several zones across the sheet. Similarly, in a calendaring stage, a segmented calendaring roller can have several actuators for controlling the nip pressure applied between the rollers at various zones across the sheet.
In a modern paper making machine, all of the actuators in a stage must be operated to maintain a uniform and high quality finished product. Such control might be attempted, for instance, by an operator who periodically monitors sensor readings and then manually adjusts each of the actuators until the desired output readings are produced. In practice, however, manual control is not fast enough for modern high-speed paper making machines, especially during an upset condition in the process. Thus, in common practice, sheetmaking machines include control systems for automatically adjusting cross-directional actuators using signals sent from scanning sensors.
In theory, if each actuator affected only one cross-directional zone of a sheet, then a sensor signal, or reading, from that particular zone could be used to control the associated actuator for that particular zone. (In the art of sheetmaking, the cross-directional zones are often referred to as "slices" because they extend along the length of the sheet in the machine direction; in the following the terms are used interchangeably.) In practice, however, each actuator not only affects its own slice on a sheet but, to a greater or lesser extent, other slices across a sheet. For example, a steam valve near the middle of a sheet might also affect the moisture content of adjacent slices. This problem is often referred to a "cross-directional coupling."
The cross-directional coupling problem can be illustrated by analogy to a simple manual control system. Consider, for example, the problem faced by the operator of a manually controlled paper making machine when only one zone has a high moisture content. Because of cross-directional coupling, the operator cannot merely increase the steam heat to the one wet zone without simultaneously reducing the steam (or adding water with a water spray) to each of the neighboring zones. But, decreasing the heat in neighboring zones will effect other zones, and so on, and so on. In other words, any corrective action in one cross-directional zone will affect the papermaking process in other zones. In control systems terminology, the dilemma faced by the machine operator can be described as a coupling of the actuator/sensor control channels for each zone.
In practice, then, the control actions of an actuator in one cross-directional zone must account for sensor measurements from each of the other zones. One general approach to solving the problem of control channel coupling is to make compensation for sensor measurements to cancel the effects of cross-directional coupling before those measurements are used to control the actuators. Conventionally, decoupling compensation is accomplished by deriving a set of mathematical relationships that describe the effect of each actuator on each of the different zones. The coupling equations can then be represented in a matrix format which is then inverted to produce a decoupling matrix. In other words, the decoupling matrix represents a set of mathematical relationships which can be used to compensate the sensor measurements in each zone for the effects of all the other actuators. Consequently, the actual sensor measurement signals from each zone can be compensated, or decoupled, and these decoupled signals can be used as independent control channels to control the paper-making machine.
One shortcoming of conventional decoupling control techniques is that the coupling matrices are usually difficult to derive. This is because the coupling matrix will have one column for each actuator and one row for each measurement slice. Consequently, for paper-making machines with many actuators and zones, the coupling matrix can be quite large, requiring substantial amount of time to derive and a significant amoumt of effort to invert. Another shortcoming of conventional decoupling control techniques is that the coupling properties, and thus the coupling matrix, may change from time to time under different operating conditions. However, conventional decoupling control techniques do not allow the decoupling matrix to be updated in a timely fashion in order to match changes in the process.
Conventional decoupling control techniques may be improved by identifying the time varying coupling matrix at different times during the process. The most recently identified coupling matrix can then be inverted and used to construct a decoupling matrix which is suitable for controlling the process at any particular time. However, a conventional control system that identified and inverted multiple coupling matrices would require a protracted amount of computation time, with the result that substantial quantities of inferior sheet material may be produced before corrective actions are implemented by the control system.
Not only is an automatic control system that involves matrix inversion too slow, but also such a system often is not accurate enough to adequately control modern high-speed sheetmaking processes. This is because calculating the inverse of a large matrix is often difficult, and sometimes impossible. In fact, even with the aid of a high speed computer, numerical matrix inversion can be very sensitive to errors in the sensor measurements which are known as instabilities in the numerical method used to invert the matrix.